From: g.scholten@gmx.de   
      
   Sabbir Rahman wrote:   
      
   >> Assumed the infalling matter crosses the event horizont *after* the   
   >> black hole has formed, i.e. when the radial coordinate R of the boundary   
   >> of the dust cloud is already < rs. Then the infalling matter first   
   >> crosses the event horizon at r = rs and then, a little later, crosses   
   >> the dust cloud boundary at r = R < rs and enters the interior of the   
   >> dust cloud. Before crossing the event horizon at rs and as well after   
   >> crossing the event horizon and before crossing the dust cloud surface at   
   >> R, the infalling matter feels the Schwarzschild metric S_{M(R)}(r,t) for   
   >> the full mass M = M(R). After crossing the dust cloud boundary at R, the   
   >> infalling matter no longers feels that Schwarzschild metric, but instead   
   >> the Schwarzschild metric S_{M(r)}(r,t) for the lower mass M(r) < M(R),   
   >> where r < R is the radial current radial coordinate of the infalling   
   >> matter. So, the infalling matter at r < R feels the same metric like the   
   >> particles of the dust cloud at r.   
   >>   
   >> Nevertheless, the infalling matter is doomed to hit the singularity,   
   >> just like the particles of the dust cloud itself are. After the dust   
   >> cloud is inside its own Schwarzschild radius, no repulsive forces can   
   >> support it against gravitational collaps any more. As we have seen, any   
   >> support against gravitational collaps that is active short before the   
   >> formation of the black hole (= short before the radial coordinate R of   
   >> the boundary of the dust cloud deceeds the Schwarzschild radius rs) is   
   >> very unstable and destroyed by the lowest disturbance.   
   >   
   > It is true that a dust cloud will continue to collapse (I have   
   > already explained why the standard picture is incorrect in this   
   > case, and what the correct picture is). However, this will not be   
   > true for a generic interior matter configuration.   
      
   You are wrong, it is true for a generic interior matter configuration,   
   too. If M is the mass of the generic interior matter configuration and   
   its boundary (surface) is located at radial coordinate R < 2GM (so that   
   the black hole has already formed), then the gravitational collaps of   
   the matter configuration is unavoidable.   
      
   Let's assume for example that the matter configuration is "rigid" in the   
   sense that there are repulsive interatomic forces that support against   
   gravitational collaps. Those repulsive forces fail to support at least   
   after R has become < 2GM.   
      
   Let's imagine the generic interior matter configuration splitted into   
   shells again. For the outermost shell at r = R(t0) < 2GM, the   
   Schwarzschild metric S_M(R(t0),t0) for the full mass M = M(R(t0))   
   applies at time t = t0. Due to that, the outermost shell collapses   
   unavoidably. If the next inner shell at R(t0) - dr does not collaps,   
   too, at time t0, the outermost shell encounters it at time t0 + dt.   
   After that, the Schwarzschild metric S_M(R(t0) - dr, t0 + dt) for the   
   full mass M applies at the next inner shell at R(t0) - dr, and by this,   
   that next inner shell is unavoidably forced to collaps, too. This shell   
   then encounters the third shell, and so on. So, shell by shell is forced   
   to collaps, yielding all shells collapsing in the end.   
      
   Or let's use another picture. Let's split the generic interior matter   
   configuration into an inner part with its boundary R_inner, let's say   
   R_inner = R(t0)/2, and an outer part R_inner < r < R(t0). The mass   
   M_inner of the inner part is much lower than the mass M of the total   
   configuration (for uniform density and without taking spatial curvature   
   into account, M_inner = M/8 would apply), so that we can assume that   
   R_inner > 2GM_inner and the inner part on its own would therfore not   
   make up a black hole. So, the inner part on its own would not be caused   
   to collaps unavoidably by its own gravity. However, the outer part is   
   caused to collaps (shell by shell), and after some time, the outer part   
   falls onto the inner part and causes it to collaps, too. So in the end,   
   the whole configuration collapses unavoidably.   
      
      
   > So if what you were saying were true in general (i.e. not specifically   
   > for a dust cloud - where as it happens it is not true either, but   
   > anyway...), then it would in principle allow a particle to fall   
   > into a black hole, and then subsequently escape from it when it   
   > finds itself in a region that is no longer inside its own Schwarzschild   
   > radius. Clearly this cannot be the case.   
      
   You are wrong, it can be the case. Let's regard the picture of inner and   
   outer part again. Since R_inner(t0) > 2GM_inner, the inner part on its   
   own does not make up a black hole currently at the time t0. In fact, we   
   can imagine a particle that enters the inner part by crossing the   
   boundary R_inner(t0) from above, and leaves it again by crossing the   
   boundary R_inner(t0) backwards from below.   
      
   However, that particle cannot move outwards very far, it cannot leave   
   the outer part outwards. And after some time, the outer part will fall   
   onto the inner part, forcing it to co-collaps. So, our particle is   
   unavoidably forced to fall back inwards again after some time, at first   
   under R_inner(t0) and finally into the singularity. Only temporarily, it   
   can escape from the inner part.   
      
   In a rotating (Kerr metric) or electrically charged (Reissner-Nordstroem   
   metric) black hole, there is indeed something similar, and not only   
   temporarily. There are two horizons, an inner one and an outer one,   
   where the outer one is comparable to the event horizon of a   
   Schwarzschild (non-rotating and uncharged) black hole. Inside the inner   
   horizon, particles can move outwards, unless they touch the inner   
   horizon. They cannot leave the inner horizon, but in the region inside   
   the inner horizon, they can move freely.   
      
      
   >>> Thus, we _do_ have two incompatible metrics here - the Schwarzschild black   
   >>> hole metric for a black hole of mass M, and the metric R(r,t) that you   
   >>> have described above, which is indeed Schwarzschild, but always for some   
   >>> mass M(R) < M for r < 2GM.   
   >>   
   >> The Schwarzschild black hole metric for a black hole of mass M applies   
   >> only outside the dust cloud, in the region r > R, that contains the   
   >> regions R < r < rs and r > rs. Inside the dust clould, only the   
   >> Schwarzschild metric for the lower mass M(r) < M applies. And at the   
   >> boundary R, both metrics match.   
   >   
   > This latter statement is incorrect for r < rs for the reasons I   
   > have already given.   
      
   As we have seen, the reason you have given are wrong.   
      
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